Exam 11: Sums of Independent Random Variables and Limit Theorems

The arm span of kindergarteners (in inches) is modelled by the random variable X, where X is N(2,.5). Find the probability that 10 kindergarteners can encircle a 21-feet tree in circumference by touching fingertips in a circle.

Suppose that all we know about the heights of trees in a redwood forest is that their average height is 162 feet with standard deviation 22. Let H be the average height of 55 trees chosen at random. Estimate P(160

For a gambling game, a person wants to estimate the percentage of the times a dart player hits the bullseye. How many darts should the player throw to be 96% sure that the estimate is within .02 of the actual percentage?

The moment generating function for a random variable is given by $M _ { X } ( t ) = \exp \left( 3 e ^ { t } - 3 \right)$ . Find P(X>2).

The moment generating function of a random variable is $M _ { X } ( t ) = \frac { 1 } { 8 } e ^ { t } + \frac { 1 } { 2 } e ^ { 4 t } + \frac { 1 } { 4 } e ^ { 5 t } + \frac { 1 } { 8 } e ^ { 7 t }$ . Find P(X=i) for 1=1,…,7.

Suppose that at a certain school, all 1500 students write letters to pen pals. The number of letters each student sends is independent of the number of letters other students send. Furthermore, the numbers of letters written by students are identically distributed random variables with mean 8 and standard deviation 2.2. Find the 90th percentile of the numbers of letters written.

At a certain Doggy Weight Loss clinic, the dogs' weights are normally distributed with mean 81 pounds and standard deviation 12 pounds. If you randomly select 12 dogs, what is the probability that the average weight of those 12 dogs is at most 80 pounds.

For a given fishing tournament, the fish caught have lengths distributed with mean 2.3 feet and standard deviation .4 feet. If you want to be 90% sure of catching the longest fish at the tournament, what should the minimum length of the longest fish you catch be?

For a random variable Y, the moment generating function is My(t)=exp(2t²+3t). Find E(Y) and σy.

X is a random variable with moment generating function $M _ { X } ( t ) = \left( \frac { 1 } { 4 } + \frac { 1 } { 4 } e ^ { t } + \frac { 1 } { 2 } e ^ { 2 } t \right) ^ { 8 }$ Find Var(3X) and P(X≤2).

Let X be a random variable with moment generating function $M _ { X } ( t ) = \frac { 4 } { \left( 2 - \frac { t ^ { 2 } } { 3 } \right) ^ { 2 } }$ for t<2. Find the moments of X.

Let the moment generating function of a random variable X be given by $M _ { X } ( t ) = \left( \frac { 3 } { 4 } e ^ { t } + \frac { 1 } { 4 } \right) ^ { 9 }$ . Find P(X>7).

Suppose that, on a random day in January, the average temperature in Duluth, MN is 0∘ F with standard deviation 4∘. Give an upper bound for the probability that a random day in January in Duluth, MN is at least 12∘.

Mary drives her car every day. The error made by her car's odometer on a random day has standard deviation 2.3 miles. Using Chebyshev's inequality, find an upper bound for the probability that after 20 days, her odometer is off by at least 15 miles.

At a certain drag race, finishing times (in seconds) for Japanese cars and American cars are N(15,2) and N(16.1,3), respectively. If 9 American cars and 9 Japanese cars race, find the probability that the average race time of the Japanese cars is at least 2 seconds faster than that of the American cars.